A Generalized Measure Concentrated on a Closed Set of Measure Zero

When E is a closed set of measure zero in the dyadic group and the Walsh series satisfies

Construction of a solution of a mixed problem for the generalized heat conduction equation by the fourier method

By the Fourier method a solution of the equation

An Elementary Approach to an Eigenvalue Estimate for Matrices

A celebrated result of Johnson, Maurey, König and Retherford from 1977 states that for every complex matrix satisfies the following eigenvalue estimate:

Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case)

We solve Tikhomirov's problem on the explicit computation of sharp constants in the Kolmogorov type inequalities

Sharp Estimates for Deviations of Linear Approximation Methods for Periodic Functions by Linear Combinations of Moduli of Continuity of Different Order

Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that is odd, . Then

Upper Bounds of the Lebesgue Constants for Fourier–Jacobi Series Summation Methods Defined by a Multiplier Function

Let be the Jacobi polynomials and let C[a,b] be the space of continuous functions on [a,b] with the uniform norm. In this paper, we study sequences of Lebesgue constants, i.e., of the norms of linear operators generated by a multiplier matrix defined by the following relations:

Sharp Kolmogorov-Type Inequalities for Moduli of Continuity and Best Approximations by Trigonometric Polynomials and Splines

In what follows, $C$ is the space of -periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm; is the mth modulus of continuity of a function f with step h and calculated with respect to P; , ( ), ,

Approximating periodic functions by interpolation sums of Jackson type

Let

Counting the number of isotone selfmappings of crowns

Letn>1. The number of all strictly increasing selfmappings of a 2n-element crown is . The number of all order-preserving selfmappings of a 2n-element crown is

On the generalized Cauchy-Riemann systems

The factorization of the Laplacian by means of first order systems and of second order operators was considered by several authors (see, e.g [2],[3],[4]). In the paper the definition of Cauchy-Riemann system (CR-system) of ordern is given by their symbols. We prove that ifD ⁽ⁿ⁾ is the symbol andD ^{(s, n)} is the sign-matrix of CR-system, then

Large deviations for martingales with some applications

Let(X _i ) be a martingale difference sequence. LetY be a standard normal random variable. We investigate the rate of uniform convergence

Analog of the Akhiezer–Krein–Favard Sums for Periodic Splines of Minimal Defect

Let be the space of 2-periodic functions whose (r – 1)th-order derivative is absolutely continuous on any segment and rth-order derivative belongs to L _p, S _2n,m is the space of 2-periodic splines of order m of minimal defect over the uniform partition . In this paper, we construct linear operators such that

Strong uniform consistency of nonparametric regression function estimates

Let (X, Y) be a ^dx-valued random vector and let r(t)=E(Y/X=t) be the regression function of Y on X that has to be estimated from a sample (X _i, Y_i), i=1,..., n. We establish conditions ensuring that an estimate of the form

Existence of Minimizers for Nonconvex Variational Problems with Slow Growth

Consider the minimization problem

Realization and Smoothness Related to the Laplacian

For f L _n (T ^d ) and , the modulus of smoothness

A Littlewood-Paley type inequality

In this note we prove the following theorem: Let u be a harmonic function in the unit ball and . Then there is a constant C = C(p, n) such that

Laplace approximations for sums of independent random vectors

Summary LetX _i,iN, be i.i.d.B-valued random variables whereB is a real separable Banach space, and a mappingB R. Under some conditions an asymptotic evaluation of is possible, up to a factor (1+o(1)). This also leads to a limit theorem for the appropriately normalized sums under the law transformed by the density exp .     

Kolmogorov-type inequalities for mixed derivatives of functions of many variables

Let = (₁,...,_d) be a vector with positive components and let D be the corresponding mixed derivative (of order _j with respect to the jth variable). In the case where d > 1 and 0 < k < r are arbitrary, we prove that

An Elementary Proof of Pólya-Vinogradov's Inequality, II

Let be a primitive character mod k, k > 2. In [1], the following elementary estimate

Behavior of Solutions of Quasilinear Elliptic Inequalities

The paper considers solutions of the coercive inequalities